Coutts, known as the bank to the rich, has been forced to compensate all customers who suffered a loss as a result of its failings in selling the AIG Life Premier Bonds.

The City watchdog said clients were told they were investing in a simple cash fund that could be seen as an alternative to a bank or building society account. In fact, a significant proportion of the fund's assets were invested in riskier asset-backed securities.

When the financial crisis of 2007 and 2008 caused the value of some of these assets to nosedive and AIG's own share price plummeted, customers started a run on the fund.

It was suspended on 15 September 2008, when 247 Coutts customers still had £748m invested in the fund, but they were allowed to withdraw only half of their investment. On 14 December 2008 customers were given the opportunity to withdraw the remaining 50% but for less than its value.

The FSA said Coutts, now owned by Royal Bank of Scotland, failed by telling investors the AIG product was a cash fund and advisers were given inadequate training about the risks and features of the product.

The bank recommended the fund to some customers even though it might have exposed them to more capital risk than they were willing to accept and many customers were advised to invest too large a proportion of their overall assets in the fund.

The bank also failed to properly deal with questions raised from December 2007 about its past sales of the fund and failed to undertake an effective compliance review of its sales of the fund after it was suspended and customers complained.

"It is imperative that firms also ensure that clients understand the nature of the product they are buying and the risks it involves. It is disappointing that Coutts failed to reflect properly upon the impact of changing market conditions and what that meant for the advice they had given, and were giving, to their customers."

Coutts has agreed to settle at an early stage in exchange for a 30% discount on its fine. Its fine would have been £9m.

Rory Tapner, chief executive of Coutts, said: "We always strive to provide the highest level of investment advice to our clients and have therefore been disappointed that we did not meet our high standards of service in respect of this product.

"We had already implemented enhancements to our investment advice procedures, which provide reassurance that the past failings identified by the FSA will not be repeated."